【作者】 崔艳芬；

【导师】 茅德康；

【作者基本信息】 上海大学 ， 计算数学， 2008， 博士

【摘要】 近几年来,茅德康等对线性传输方程设计了一种能保持两个和三个离散守恒律的差分格式(见[44],[45],[15],[48]和[49]),其数值效果无论在解的精度还是长时间的数值模拟方面都远胜于传统的差分格式。本文的第一个工作是对线性传输方程的保持两个守恒律的差分格式进行了数值分析,揭示了这种格式在计算中各步的误差会相互抵消这一性质。这种性质在目前我们所见过的数值方法中是罕见的。正是因为格式的这种性质,它能远胜于传统的格式。本文的第二个工作是将这种设计满足线性传输方程多个守恒律的差分方法应用于KdV方程,对之设计了满足两个守恒律的差分格式。具体的做法是将KdV方程分裂成守恒方程部分和散射方程部分,而我们的方法是应用在守恒方程部分的离散上。所设计的格式在长时间的数值模拟中表现出十分优秀的品质。更多还原

【Abstract】 In recent years, Mao and his co-workers developed difference schemes for linear advection equation which satisfied two or three discrete conservation laws, see [44], [45], [15], [48] and [49]. The numerical results of the developed schemes were far better than traditional difference schemes’ at both solutions’ accuracy and long-time numerical integrations.The first work in this paper is to do the numerical analysis for the difference scheme staisfing two discrete conservation laws for the linear advection equation. We reveal that numerical errors of the scheme at successive time steps cancel with each other, which is a feature that is rarely seen in the numerical methods we have ever known. It is this feature of error-self-canceling that makes our scheme far better than traditional schemes.The second work in this paper is to apply the numerical approach developed for the linear advection equation to the KdV equation. We develop a scheme for the KdV equation, which satisfies the first two conservation laws of the equation. In constructing the scheme, we adopt the splitting strategy to split the equation into the conservation part and the dispersion part, and the approach of designing scheme satisfying two conservation laws is applied in the discretization of the conservation part. The developed scheme shows good quality in long-time numerical simulations.更多还原